Examining the Effects of Traders’
Overconfidence on Market Behavior
Chia-Hsuan Yeh a,∗ , Chun-Yi Yang b
a Department
b Tinbergen
of Information Management, Yuan Ze University, Chungli, Taoyuan
320, Taiwan
Institute, Roetersstraat 31, 1018 WB Amsterdam, The Netherlands
Abstract
This paper provides an agent-based artificial financial market to examine the
effects of traders’ overconfidence. Traders’ overconfidence is presented in the way
that they usually underestimate their volatility. We find that overconfidence results
in higher volatility, price distortion, and trading volume. The phenomena of fat-tail
of return distribution and volatility clustering are more significant when traders are
overconfident.
Key words: Rationality, Overconfidence, Artificial Stock Market, Agent-Based
Modeling, Genetic Programming
JEL classification: D83, D84, G11, G12
1
Introduction
It is well-known that modern financial economic theory heavily relies on the assumption that
the representative agent in the market behaves rationally and has rational expectations. Under this assumption, it is shown that asset prices fully reflect all available information and
always reflect their intrinsic value. In this situation, future price movements cannot be predicted on the basis of past information. Any financial regulation imposed on the market
should generate no substantial effects but result in delayed revelation of the information.
Milton Friedman is one of the strongest advocates for supporting rational expectations approach. In Friedman (1953, p. 175): “People who argue that speculation is generally destabilizing seldom realize that this is largely equivalent to saying that speculators lose money,
since speculation can be destabilizing in general only if speculators on the average sell when
the currency is low in price and buy when it is high.”
∗ Corresponding author. Tel: +886 3 4638800 ext. 2616, Fax: +886 3 4352077
Email addresses: [email protected] (Chia-Hsuan Yeh), [email protected]
(Chun-Yi Yang).
URL: http://comp-int.mis.yzu.edu.tw/faculty/yeh/ (Chia-Hsuan Yeh).
1
Examining the efficiency of real financial markets has been an interesting topic in the past
three decades. Many studies have questioned the validity of the Efficient Market Hypothesis (EMH) in real financial markets and provide the theoretical foundations or empirical
evidences to show the existence of market inefficiency. De Long et al. (1991) points out
that noise traders may survive in the long run and exert their impact on price dynamics.
Kogan et al. (2006) further indicate that irrational traders can persistently maintain a large
impact even though their relative wealth becomes quite small. In Lo and MacKinlay (1988),
Campbell and Shiller (1988), Brock et al. (1992), and Neely et al. (1997), they all find the
evidence of predictability and profitability in financial markets. In addition, the increasing
empirical evidences have indicated that traditional asset pricing models such as the capital
asset pricing model (CAPM), arbitrage pricing theory (APT), and intertemporal capital asset pricing model are unable to provide explanations regarding the stylized facts. Financial
markets usually experience several anomalies such as event-based return predictability, shortterm momentum, long-term reversal, high volatility of asset prices relative to fundamentals
where bubbles and crashes never cease. These phenomena cannot be purely explained by the
changes in fundamentals. Given these findings, it is reasonable to reexamine the theory of
finance based on imperfect rationality.
Actually, studying economics and finance from the perspective of imperfective rationality
has both theoretical and empirical foundations. The reason for economists holding the assumption of rational expectations is that the economic systems without this restriction may
produce numerous outcomes so that prediction is impossible. Simon (1957) argued that
agents possess imperfect information or knowledge regarding the environment and they also
have limited ability in processing information. Therefore, bounded rationality is a more reasonable and more appropriate description regarding agents’ behavior than perfect rationality.
The empirical evidence from cognitive psychology also supports that agents do not behave
rationally. As mentioned in Hirshleifer (2001), traders in financial markets exhibit several
phenomena deviated from perfect rationality such as overreaction toward salient news, underreaction toward less salient news, anchoring, loss aversion, mental accounting, herding,
and the overconfidence.
In the past two decades, research studies devoted in financial economics have considered the
models deviated from full rationality. One branch focuses on the effects of noise traders, e.g.
De Long et al. (1989, 1990a, 1990b, 1991) and Shleifer and Summers (1990). Their finding
have demonstrated that the presence of noise traders can generate substantial effects which
are quite different from those observed in the market populated with rational traders alone.
The other branch focuses on the consequences resulting from traders’ psychological biases.
This line of research has been an important issue in the field of behavioral finance.
DeBondt and Thaler (1995) states that perhaps the most robust finding in the psychology
of judgment is that people are overconfident. In Tversky and Kahneman (1974), they points
out that traders’ overconfidence may be due to an “anchoring and adjustment” process.
The anchor has a major influence so that the adjustment is usually insufficiently. Therefore,
traders have tight subjective probability distributions. This phenomenon is also evidenced by
empirical literature on judgment under uncertainty. Benos (1998) then believes that selection
and survivorship biases may also be sources of overconfidence and successful traders usually
2
overestimate their own contribution to their success. Such a reasoning is supported by the
attribution theory, e.g. Bem (1965), in which it describes that individuals usually attribute
outcomes that support the validity of their decisions to high ability, and outcomes that are
inconsistent with the decisions to external noise.
However, theoretical results heavily rely on specific assumptions regarding the characteristics
of traders as well as the market environments, and the information structures. Since many
factors are involved, and traders’ behavior may generate externalities on others, it would
have a more clear and concrete picture about the effects of traders’ psychological biases if a
heterogeneous-agent framework is employed. In fact, Hirshleifer (2001) mentioned that,
The great missing chapter in asset-pricing theory, I believe, is a model of the social process
by which people form and transmit ideas about markets and securities. (p. 1577)
Under a well-controlled heterogeneous-agent environment where traders’ psychological factors are considered, we are able to examine the market phenomena from the perspective
of micro-foundation. However, such a framework would be too complicated so that analytical results are difficult to be derived. Therefore, a simulated framework composed of many
heterogeneous and bounded-rational traders whose learning behavior is appropriately represented would be a better architecture. In this paper, we provide an agent-based artificial
financial market to examine the effects of traders’ overconfidence on several stylized facts
such as volatility clustering, and fat tails for the return series.
The remainder of this paper is organized as follows. The basic framework of the model which
includes the market environment, the traders’ learning behavior, and the mechanism of price
determination are described in Section 2. Section 3 presents the simulation design, and the
results are summarized in Section 4. Section 5 concludes.
2
2.1
The model
Market structure
The basic framework of the artificial stock market considered in this paper is the standard asset pricing model with many heterogeneous traders. All traders are characterized by bounded
rationality in which they are equipped with adaptive learning behavior represented by the
genetic programming (GP) algorithm. In the framework of GP, traders are freely allowed to
form various types of forecasting functions which may be fundamental-like or technical-like
rules in different time periods
Our framework is very similar to that used in Yeh (2008). However, to calibrate the model
being able to fit different time horizons of real financial markets, we follow the design proposed in He and Li (2007).
Consider an economy with two assets. One is the risk free asset called money which is perfectly elastically supplied. Its gross return is R = 1 + r/K, where r is a constant interest rate
per annum and K represents the trading frequency measured in one year. For example, K=1,
12, 52, and 250 stand for the trading periods of year, month, week, and day, respectively.
The other asset is a stock with a stochastic dividend process (Dt ) not known to traders. The
3
trader i’s wealth at t + 1, Wi,t+1 , is given by
Wi,t+1 = RWi,t + (Pt+1 + Dt+1 − RPt )hi,t ,
(1)
where Pt is the price (ex dividend) per share of the stock and hi,t denotes the shares of the
stock held by trader i at time t. Let Rt+1 be the excess return at t + 1, i.e. Pt+1 + Dt+1 − RPt ,
and Ei,t (·) and Vi,t (·) are the forecasts of trader i about conditional expectation and variance
at t + 1 given his information up to t (the information set Ii,t ), respectively. Then we have
Ei,t (Wt+1 ) = RWi,t + Ei,t (Pt+1 + Dt+1 − RPt )hi,t = RWi,t + Ei,t (Rt+1 )hi,t ,
2
Vi,t (Wt+1 ) = h2i,t Vi,t (Pt+1 + Dt+1 − RPt ) = h2i,t Vi,t (Rt+1 ) = h2i,t σi,t
,
(2)
(3)
2
is the conditional variance of (P + D) given Ii,t . Assume that all traders follow the
where σi,t
same constant absolute risk aversion (CARA) utility function, i.e. U (Wi,t ) = −exp(−λWi,t ),
where λ is the degree of absolute risk aversion. At the beginning of each period, each trader
myopically maximizes the one-period expected utility function subject to Eq. (1). Assuming
that traders’ forecasts regarding the sum of the next period’s price and dividend follow the
Gaussian distributions, trader i’s optimal share of stock holding, h∗i,t , solves
λ
max{Ei,t (Wt+1 ) − Vi,t (Wt+1 )},
h
2
(4)
that is,
h∗i,t =
Ei,t (Rt+1 )
.
λVi,t (Rt+1 )
(5)
If it is supposed that the current stock holding for trader i is at the optimal level, i.e.
h∗i,t = hi,t , then the trader’s reservation price, PiR , can be derived.
PiR =
2
Ei,t (Pt+1 + Dt+1 ) − λhi,t σi,t
.
R
(6)
When traders possess homogeneous rational expectations, the fundamental price Pf is given
by
Pf,t =
∞
∑
Et [Dt+i ]
i=1
Ri
.
(7)
2.2 Learning of traders
According to Eq. (6), it is shown that traders’ reservation prices rely on their conditional
expectations and variances regarding P + D. We adopt the functional form for Ei,t (·):


 (Pt + Dt )[1 + θ0 tanh( ln(1+fi,t ) )]
ω
Ei,t (Pt+1 + Dt+1 ) = 
 (P + D )[1 −
t
t
i,t |)
θ0 tanh( ln(|−1+f
)]
ω
4
if fi,t ≥ 0.0,
if fi,t < 0.0,
(8)
where fi,t is evolved using GP based on Ii,t . 1
The modeling of traders’ conditional variances also plays an important role. Here we consider
the following form of conditional variance:
2
2
σi,t
= (1 − θ1 − θ2 )σi,t−1
+ θ1 (Pt + Dt − ut−1 )2 + θ2 [(Pt + Dt ) − Ei,t−1 (Pt + Dt )]2 ,
(9)
where
ut = (1 − θ1 )ut−1 + θ1 (Pt + Dt ).
(10)
Traders update their own estimated conditional variance of the active rule at the end of each
period.
Each trader’s overconfidence level is modeled as the degree of underestimation about the
2
, is replaced
conditional variance. Therefore, the conditional variance shown in Eq. (9), σi,t
2
by Ωi,t :
2
Ω2i,t = γ(t)σi,t
,
(11)
where




γ1 ,



γ(t) =
γ2 ,





 1,
if profit > 0,
(12)
if profit < 0,
if profit = 0.
The values of γ1 (γ2 ) should be smaller (greater) than 1, and |1 − γ1 | > |γ2 − 1|. The last
condition is used to model the behavior of biased self-attribution.
Each trader possesses several models, say NI , which are represented by GP. The performance
of each forecasting model is indicated by the value of strength which is defined by
si,j,t = −Ω2i,j,t ,
(13)
where si,j,t is the strength of the jth model for trader i in period t. Traders learn to make
better forecasts through an adaptation process that abandons the model with poorest performance and generates a new one by means of an evolutionary process devised in GP. The
evolutionary process takes place every NEC periods (evolution cycle) for each trader asynchronously. Traders’ learning works as follows. At the beginning of each evolution cycle, each
trader randomly chooses NT out of NI models. The one with the highest strength value is
1
Regarding the formation of a function by means of GP as well as the implementation of GP, the
reader should refer to Yeh (2007).
5
selected as the model he uses in these periods of this evolution cycle. At the end of each
evolution cycle, the model with the lowest strength is replaced by the model which is created
by means of crossover, mutation, or immigration.
A simplified double auction (DA) is employed as the trading mechanism. Each period is
decomposed into NR rounds. At the beginning of each round, a new random permutation
of all traders is performed to determine their order of bid and ask. Each trader, based on
his own reservation price and current best bid or ask, he makes a decision of his offer. If a
bid (ask) exists, any subsequent bid (ask) must be higher (lower) than the current one. For
the sake of simplicity, only a fixed amount of stock (∆h) is traded in each transaction. The
last transaction price (closing price) in each period is recorded as the market price for this
period. 2
3
Simulation design
3.1 Stylized facts
Before conducting our simulations, we first present several stylized facts observed in real
financial markets. The basic statistical properties of the Dow Jones Industrial Average Index
(DJIA), Nasdaq Composite Index, and S&P 500 are summarized in Table 1.
The third and fourth columns show the minimum and maximum returns in percentage,
respectively. The market volatility in terms of the average of absolute returns is described
in the fifth column. The sixth column is the kurtosis K. It is evident that the kurtosis of
all markets are greater than 3, which is an indication of fat tails. The tail index α which
is a more reliable estimator of fat-tail is presented in seventh column. The smaller the α
value is, the fatter the tail is. The α value is obtained based on 5 percent of the largest
observations. Hurst exponent is employed to identify whether a time series follows a random
walk or it possesses underlying trends. The value of Hurst exponent (H) is between 0 and
1. A random series has the value of 0.5, while 0.5 < H < 1 (0 < H < 0.5) implies a time
series with persistence (anti-persistence). The Hurst exponents of raw returns and absolute
returns are shown in the last two columns, respectively. In Table 1, the raw return series
of all markets are close to random series. By contrast, the absolute return series exhibit
strong sign of volatility clustering. These phenomena can be also evidenced in Fig. 1 which
displays basic properties of Nasdaq. Fig. 1 is the time series plot during 1972-2007. The
distribution of the returns is presented in the third panel of Fig. 1 in which the black line
is the normal distribution with the same variance. It is clear that the return distribution
of Nasdaq possesses higher probabilities around the mean and the tails than those of the
normal distribution. In addition, at the 5% significance level, the insignificant autocorrelation
of raw returns for most lag periods and the significant autocorrelation of absolute returns
are exhibited in the last two panels.
3.2 Model calibration
We calibrate the model to mimic the stylized facts of the daily data in financial markets
observed in Table 1 and Fig. 1. The model parameters are shown in Table 2. Adopting
the same setup of He and Li (2007), the annual interest rate r is set as 5%, i.e., the daily
interest rate, rd = 0.05/250 = 0.02%. The daily dividend process is assumed to follow normal
2
For a more detailed implementation, please refer to Yeh (2008).
6
Table 1
Stylized facts of financial markets
Series
Period
rmin rmax
|r|
K
α
Hr
H|r|
1972-2007
-29.22
9.21
0.72
77.03
4.25
0.53
0.96
Nasdaq
1972-2007
-12.80
12.41
0.79
13.53
3.39
0.57
0.97
S&P 500
1972-2007
-25.73
8.34
0.70
53.96
4.69
0.53
0.97
NASDAQ (1,000)
DJIA
5
4
3
2
1
0
1972
1977
1982
1987
1992
1997
2002
2007
1997
2002
2007
Return (1/100)
Year
10
5
0
−5
−10
1972
1977
1982
1987
1992
Prob.density
Year
50
40
30
20
10
0
−0.10
−0.05
0.00
0.05
0.10
Return
Autocorrelation of r
0.2
0.1
0.0
−0.1
−0.2
0
20
40
60
80
100
60
80
100
Autocorrelation of lrl
Lag
0.4
0.3
0.2
0.1
0.0
0
20
40
Lag
Fig. 1. Time series properties of Nasdaq
2
= 0.004. Therefore, the fundamental price
distribution with mean D̄ = 0.02 and variance σD
7
Table 2
Parameters for simulations
The stock market
Shares of the stock (h) for each trader
1
Initial money supply for each trader
$100
Interest rate (r, rd )
(0.05, 0.0002)
Stochastic process (Dt )
2 )=N(0.02, 0.004)
N(D̄, σD
Amount for each trade (∆h)
1
Maximum shares of stock holding
10
Number of rounds for each period (NR )
50
Number of periods (NP )
20,000
Traders
Number of traders (N )
100
Number of strategies for each trader (NI )
20
Tournament size (NT )
5
Evolution cycle (NEC )
5
λ
0.5
θ0
0.5
ω
15
θ1
0.01
θ2
0.001
Parameters of genetic programming
Function set
{if-then-else;and,or,not;≥,≤,=
+,-,×,%,sin,cos,abs,sqrt}
Terminal set
{Pt−1 , ..., Pt−5 , Dt−1 , ..., Dt−5 }
Selection scheme
Tournament selection
Tournament size
2
Probability of creating a tree by immigration
0.1
Probability of creating a tree by crossover
0.7
Probability of creating a tree by mutation
0.2
of our model is
Pf =
∞
∑
D̄
i=1
Ri
=
D̄
= 100.
rd
(14)
Short selling and buying on margin are prohibited. Each trader utilizes the information regarding the stock price and dividend history up to the last 5 periods to form his expectations.
Table 3 summarizes the basic statistical properties for 20 simulations and Fig. 2 display
the time series properties of a typical run. In comparison with the results obtained in real
financial markets, our model fits these stylized facts very well. The fifth, seventh, and eighth
8
Table 3
Statistical properties of the calibrated model
rmin rmax
|r|
PD
K
V
σV
α
Hr
H|r|
Minimum
-13.85
11.80
0.31
25.21
18.57
157.18
18.50
1.91
0.47
0.90
Median
-20.11
17.32
0.49
43.39
49.23
168.26
21.15
3.41
0.52
0.91
Average
-23.38
20.76
0.50
43.91
45.33
167.81
21.56
3.38
0.52
0.92
Maximum
-45.11
32.10
0.78
66.71
81.11
173.70
26.13
4.96
0.57
0.94
V
σV
α
Hr
H|r|
Table 4
Statistical properties of the model with overconfident traders
rmin
rmax
|r|
PD
K
Minimum
-16.08
19.23
0.73
60.15
9.47
161.15
30.36
1.32
0.50
0.82
Median
-47.30
47.24
1.90
98.43
23.54
177.11
49.73
3.68
0.56
0.95
Average
-57.95
963.36
5.97
97.17
209.09
179.81
47.52
3.60
0.59
0.93
Maximum
-99.81
5400.00
32.97
123.32
1128.09
222.91
55.00
7.10
0.77
0.97
columns of Table 3 are the price distortion (PD ), the trading volume and its standard deviation, respectively. Price distortion which measures the degree of price deviation from the
fundamental price is defined as
PD =
4
NP
100 ∑
Pt − Pf
|
|
NP t=1
Pf
(15)
Simulation results
Based on the parameters shown in Table 2, we examine the consequences of overconfident
traders. Traders’ overconfidence is represented by the way that they underestimate their
conditional variances. Each trader’s overconfidence level is determined by two parameters,
γ1 and γ2 . In this paper, we choose γ1 = 0.99 and γ2 = 1.005. The results of 20 simulation
runs are presented in Table 4, and the time series properties of a typical run is plotted in
Fig. 3.
For most of runs, our simulated market with overconfident traders still provides a good fit
of the stylized facts. For example, return distribution displays the property of fat-tail. The
autocorrelation of raw return series is insignificant and that for the absolute returns is quite
significant. Overconfidence makes market exhibit richer dynamics and stronger characteristics. First, price dynamics is more volatile and the scale of bubble and crash is larger. It is
no doubt that price distortion would be more serious. In the market without overconfident
traders, the median of market volatility is 0.49%. By contrast, it is 1.90% in the market
composed of overconfident traders. Second, comparing the second panel of Figs. 2 and 3,
it is clear that overconfidence results in larger return variation. In Table 3, the median of
the minimum (maximum) of returns is -20.11% (17.32%) among 20 runs, while it is -47.30%
(47.24%) when traders are overconfident. Overconfidence also causes more significant volatility clustering. This can be evidenced from the higher autocorrelation of absolute returns.
Third, from Tables 3 and 4, we observe that trading volume as well as its volatility increase
when traders are overconfident. Basically, our findings confirm the analytical results ob9
Price
120
100
80
60
40
20
0
0
5000
10000
15000
20000
0
5000
10000
15000
20000
Period
Prob.density
Return (1/100)
30
20
10
0
−10
−20
−30
Period
60
40
20
0
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05
0
0.05 0.1 0.15 0.2 0.25 0.3
Return
Autocorrelation of r
0.2
0.1
0.0
−0.1
−0.2
0
20
40
0
20
40
Lag
60
80
100
60
80
100
Autocorrelation of lrl
0.4
0.3
0.2
0.1
0.0
Lag
Fig. 2. Time series properties of the calibrated model
tained in Benos (1998) and Odean (1998) where they conclude that overconfidence results
in increased price volatility and trading volume.
5
Conclusion
How traders’ psychological factors affect market properties has been paid much attention in
the past decade. Overconfidence is one of most important characteristics. Under an agentbased framework, this paper examines the effects of traders’ overconfidence on the market.
The preliminary results have shown that overconfidence increases market volatility, price
distortion, and trading volume. Some stylized facts such as fat-tail of return distribution
and volatility clustering would be more evident. Of course, the results crucially rely on the
10
300
Price
200
100
0
0
5000
10000
15000
20000
0
5000
10000
15000
20000
Period
Prob.density
Return (1/100)
30
20
10
0
−10
−20
50
40
30
20
10
0
−0.2
Period
−0.1
0.0
0
20
40
0
20
40
Return
0.1
0.2
0.3
Autocorrelation of r
0.2
0.1
0.0
−0.1
−0.2
Lag
60
80
100
60
80
100
Autocorrelation of lrl
0.4
0.3
0.2
0.1
0.0
Lag
Fig. 3. Time series properties of the model with overconfident traders
design of traders’ overconfident behavior. Further investigation regarding this issue would
be necessary in the following studies.
Acknowledgement
Research support from NSC Grant No. 98-2410-H-155-021 is gratefully acknowledged.
References
Bem, D. J., 1965. An experimental analysis of self-persuasion. Journal of Experimental Social
Psychology 1, 199-218.
11
Benos, A. V., 1998. Aggressiveness and survival of overconfident traders. Journal of Financial
Markets 1, 353-383.
Brock, W., Lakonishok, J., LeBaron, B., 1992. Simple technical trading rules and the stochastic properties of stock returns. The Journal of Finance 47, 1731-1764.
Campbell, J. Y., Shiller, R., 1988. The dividend-price ratio and expectations of future dividends and discount factors. The Review of Financial Studies 1, 195-227.
DeBondt, W. F. M., Thaler, R. H., 1995. Financial decision-making in markets and firms: a
behavioral perspective. In: Jarrow, R. A., Maksimovic, V., Ziemba, W. T. (Eds), Finance,
Handbooks in Operations Research and Management Science 9, 385-410. North Holland,
Amsterdam.
De Long, J. B., Shleifer, A., Summers, L. H., Waldmann, R. J., 1989. The size and incidence
of the losses from noise trading. The Journal of Finance 44, 681-696.
De Long, J. B., Shleifer, A., Summers, L. H., Waldmann, R. J., 1990a. Positive feedback
investment strategies and destabilizing rational speculation. The Journal of Finance 45,
379-395.
De Long, J. B., Shleifer, A., Summers, L. H., Waldmann, R. J., 1990b. Noise trader risk in
financial markets. The Journal of Political Economy 98, 703-738.
De Long, J. B., Shleifer, A., Summers, L. H., Waldmann, R. J., 1991. The survival of noise
traders in financial markets. The Journal of Business 64, 1-19.
Friedman, M., 1953. The case for flexible exchange rates. In: Essays in Positive Economics.
University of Chicago Press, Chicago, IL.
He, X.-Z., Li, Y., 2007. Power-law behaviour, heterogeneity, and trend chasing. Journal of
Economic Dynamics and Control 31, 3396-3426.
Hirshleifer, J., 2001. Investor psychology and asset pricing. The Journal of Finance 56, 15331597.
Kogan, L., Ross, S. A., Wang, J., Westerfield, M. M., 2006. The price impact and survival
of irrational traders. The Journal of Finance 61, 195-229.
Lo, A. W., MacKinlay, A. C., 1988. Stock prices do not follow random walks: evidence from
a simple specification test. The Review of Financial Studies 1, 41-66.
Neely, C., Weller, P., Dittmar, R., 1997. Is technical analysis in the foreign exchange market profitable? a genetic programming approach. Journal of Financial and Quantitative
Analysis 32, 405-426.
Odean, T., 1998. Volume, volatility, price, and profit when all traders are above average.
The Journal of Finance 53, 1887-1934.
Shleifer, A., Summers, L. H., 1990. The noise trader approach to finance. The Journal of
Finance 4, 19-33.
Simon, H. A., 1957. Models of Man. Wiley, New York, NY.
Tversky, A., Kahneman, D., 1974. Judgement under uncertainty: heuristics and biases. Science 185, 1124-1131.
Yeh, C.-H., 2007. The effects of intelligence on price discovery and market efficiency. Computational Economics 30, 95-123.
Yeh, C.-H., 2008. The role of intelligence in time series properties. Journal of Economic
Behavior and Organization 68, 613-625.
12